Optimal. Leaf size=48 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}} \]
[Out]
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Rubi [A] time = 0.124061, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[3 - 2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 13.7575, size = 49, normalized size = 1.02 \[ \frac{\sqrt [4]{6} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | -1\right )}{2} - \frac{\sqrt [4]{6} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | -1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(-2*x**4+3)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0521902, size = 38, normalized size = 0.79 \[ \frac{\sqrt [4]{3} \left (E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[3 - 2*x^4],x]
[Out]
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Maple [A] time = 0.082, size = 69, normalized size = 1.4 \[ -{\frac{\sqrt{3}\sqrt [4]{6}}{18}\sqrt{9-3\,{x}^{2}\sqrt{6}}\sqrt{9+3\,{x}^{2}\sqrt{6}} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt [4]{6}}{3}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt [4]{6}}{3}},i \right ) \right ){\frac{1}{\sqrt{-2\,{x}^{4}+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(-2*x^4+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.92152, size = 39, normalized size = 0.81 \[ \frac{\sqrt{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(-2*x**4+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="giac")
[Out]